Download On the Homology of the Ginzburg Algebra Stephen Hermes

On the Homology of the Ginzburg Algebra
Stephen Hermes
Brandeis University, Waltham, MA
Maurice Auslander Distinguished Lectures
and International Conference
Woodshole, MA
April 23, 2013
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
1 / 18
Outline:
1
Ginzburg Algebra of a QP
2
Relation with the Preprojective Algebra
3
A∞ -Algebras
4
Application to the Ginzburg Algebra
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
2 / 18
Quivers with Potential
Definition (Derksen-Weyman-Zelevinsky)
A quiver with potential (QP for short) is a pair (Q, W ) where Q is a
quiver with
no loops
no 2-cycles
and W is a potential on Q, i.e., an element of
HH0 (kQ) = kQ/[kQ, kQ].
Equivalently, W is a linear combination of cycles of Q considered up to
cyclic equivalence.
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
3 / 18
The Ginzburg Algebra
The Ginzburg algebra Γ(Q,W ) of a QP (Q, W ) is the dga constructed as
b where Q
b is the quiver:
follows. As a graded algebra, Γ(Q,W ) = k Q
1
Start with Q (in degree 0).
2
Add reversed arrows α∗ : j → i (degree −1) for each α : i → j in Q.
3
Add loops ti (degree −2) for each vertex i of Q.
Example
t1
1
α
/2
β
/3
t2
α (
1h
α∗
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
t3
β
2h
(
3
β∗
April 23, 2013
4 / 18
The Ginzburg Algebra
Equipped with a differential d determined by:
dα = 0 for α ∈ Q1
dα∗ = ∂α W for α ∈ Q1
where ∂α : HH0 (kQ) → kQ (the cyclic partial derivative) given by
X
∂α (w ) =
vu
w =uαv
[α, α∗ ]
P
dti = ei
ei
α∈Q1
([x, y ] = xy − yx).
Extend to all of Γ(Q,W ) by Leibniz law:
d(xy ) = d(x)y + (−1)|x| xd(y ).
Example
t1
t2
α
1h
α∗
(
t3
β
2h
β∗
Stephen Hermes (Brandeis University)
(
3
dα = dβ = dα∗ = dβ ∗ = 0
dt1 = αα∗ , dt3 = −β ∗ β
dt2 = ββ ∗ − α∗ α
On the Homology of the Ginzburg Algebra
April 23, 2013
5 / 18
The Ginzburg Algebra
If Q is acyclic (e.g. Q Dynkin) HH0 (kQ) = 0; hence the only potential Q
admits is the trivial one W = 0. In this situation we write ΓQ = Γ(Q,0) .
For Q acyclic kQ = H 0 ΓQ . But what about higher degrees?
Definition
b to be the number of times γ traverses
Define the weight of a path γ in Q
a loop ti .
Gives a weight grading on ΓQ . Descends to a grading on H ∗ ΓQ . Denote
the weight w component of H ∗ ΓQ by Hw∗ ΓQ .
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
6 / 18
The Preprojective Algebra
Recall the preprojective algebra of Q is the algebra ΠQ = kQ/(ρ) where
b consisting of arrows of weight 0.
Q is the subquiver of Q
P
ρ = α∈Q1 [α, α∗ ].
Example
1
α
/2
β
/3
α
1h
α∗
(
β
2h
(
3
β∗
ΠQ contains kQ as a subalgebra. As a (right) kQ-module it splits into a
direct sum of preprojective indecomposable modules with each isoclass
represented exactly once.
ΠQ = H0∗ ΓQ ⊂ H ∗ ΓQ . This inclusion is proper in general.
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
7 / 18
The Preprojective Algebra
For Q Dynkin, we have an (covariant) involution η of ΠQ :
1
Let η̄ be the involution of the underlying graph |Q| of Q
(
identity map
|Q| = D2n , E7 , E8
η̄ =
unique non-trivial involution |Q| = An , D2n+1 , E6
2
This determines an involution of Q by requiring η(α) : η(i) → η(j) for
α : i → j in Q.
3
Determines an involution of kQ preserving (ρ) and so gives an
involution η of ΠQ .
Example
1
3
RRRR
2
llll
η
···
Stephen Hermes (Brandeis University)
(2n + 1)
#
3
1
RRRR
2
llll
On the Homology of the Ginzburg Algebra
···
(2n + 1)
April 23, 2013
8 / 18
The Preprojective Algebra
The involution η is used to construct D b (kQ) from mod kQ:
Example
Q:1→2→3
P3
? P2



P1 = I 3
x< EEE
x
x
EE
xx
"
< I2
FF
y
y
FF
F" yyy
Stephen Hermes (Brandeis University)
S2
P3 [1]
HH η(1)
}>
HH
}
}
H$
}
}
P
[1]
AA
: P2 [1] KKK η(2)
AA
vv
K
v
K
v
K
A vv
%
P
[1]
I1
On the Homology of the Ginzburg Algebra
P1 [1]
Pη(3) [1]
April 23, 2013
9 / 18
The Homology of ΓQ
Theorem (H.)
Suppose Q is Dynkin. Then there is an algebra isomorphism
η
H ∗ ΓQ ∼
= ΠQ [u]
where ΠηQ [u] is the η-twisted polynomial algebra. Moreover, under this
isomorphism, polynomial degree corresponds to weight.
As a k-vector space ΠηQ [u] = ΠQ ⊗k k[u]; the multiplication is given by
(xu p ) · (yu q ) = xη p (y )u p+q
for x, y ∈ ΠQ .
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
10 / 18
Remark on the Proof
The proof is given by showing both H ∗ ΓQ and ΠηQ [u] are isomorphic to
M
H ∗ Pdg (kQ)(kQ, τ −n kQ)
n≥0
where Pdg (kQ) denotes the dg category of bounded projective complexes
of kQ-modules with morphisms of arbitrary degree, and τ denotes the
Auslander-Reiten translate.
The element u ∈ ΠηQ [u] comes from the evident map kQ → kQ[1].
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
11 / 18
The Homology of Γ(Q)
Corollary (Folklore?)
There is an isomorphism
H ∗ ΓQ ∼
=
M
F n kQ
n≥0
in D b (kQ) where F = τ − [1].
Knowing H ∗ ΓQ is nice, but we really want ΓQ . To recapture ΓQ we need
to know an A∞ -structure on H ∗ ΓQ .
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
12 / 18
A∞ -Algebras
Definition
An A∞ -algebra is a k-module V together with “multiplications”
µn : V ⊗n → V ,
n≥1
satisfying the relations
X
(−1)p+qr µp+1+r ◦ 1⊗p ⊗ µq ⊗ 1⊗r = 0.
n=p+q+r
q≥1,p,r ≥0
n=1: µ1 ◦ µ1 = 0 i.e., (V , µ1 ) is a chain complex
n=2: µ2 ◦ (1 ⊗ µ1 + µ1 ⊗ 1) = µ1 ◦ µ2
i.e., µ2 : V ⊗ V → V is a chain map.
n=3: µ2 ◦ (µ2 ⊗ 1) − µ2 ◦ (1 ⊗ µ2 ) = µ1 ◦ µ3 + µ3 ◦ dV ⊗3
i.e., µ2 is associative up to a homotopy µ3
n=4: . . .
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
13 / 18
Examples
Any ordinary associative algebra (µn = 0 for n 6= 2); Conversely, for
an A∞ -algebra V with µ1 = 0, (V , µ2 ) is an associative algebra.
Any differential graded algebra (µn = 0 for n ≥ 2)
Any chain complex homotopy equivalent to an A∞ -algebra (not true
for dgas!)
Remark
Two dgas (A, dA , µA ) and (B, dB , µB ) are quasi-isomorphic (as dgas) if
and only if the A∞ -algebras (A, dA , µA , 0, . . . ) and (B, dB , µB , 0, . . . ) are
quasi-isomorphic (as A∞ -algebras).
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
14 / 18
Kadeishvili’s Theorem
Theorem (Kadeishvili)
Let A be a dga. Then there is a unique A∞ -algebra (H ∗ A, µ1 , µ2 , . . . ) so
that:
µ1 = 0
µ2 is the usual multiplication
the map j : HA → A given by choosing representative cycles is a
quasi-isomorphism of A∞ -algebras.
The A∞ -algebra H ∗ A above is called the minimal model of A.
Kadeishvili’s Theorem says dgas are determined (up to quiso) by their
minimal models (up to A∞ -quiso).
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
15 / 18
The Minimal for ΓQ
η
Recall there is an isomorphism H ∗ ΓQ ∼
= ΠQ [u].
Theorem (H.)
Suppose Q Dynkin and let (H ∗ ΓQ , 0, µ2 , µ3 , . . . ) be the minimal model of
ΓQ .
2
The maps µn are u-equivariant.
The element u ∈ µ3 Π⊗3
and so H ∗ ΓQ is generated as an
Q
A∞ -algebra by ΠQ .
3
The higher multiplications µn = 0 for n > 3.
1
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
16 / 18
Remark on Proofs
Recall
H ∗ ΓQ ∼
=
M
H ∗ Pdg (kQ)(kQ, τ −n kQ)
n≥0
and u maps to kQ → kQ[1] under this isomorphism.
The category Pdg (kQ) of projective complexes is a dg category so
H ∗ Pdg (kQ) is an A∞ -category. Transfers to A∞ -structure on
L
H ∗ Pdg (kQ)(kQ, τ −n kQ).
Proofs given by studying A∞ -structure on H ∗ Pdg (kQ).
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
17 / 18
Thank You!
Stephen Hermes (Brandeis University)
On the Homology of the Ginzburg Algebra
April 23, 2013
18 / 18